Quadratic invariants of the elasticity tensor

arXiv:1509.02315v1 [cond-mat.other] 8 Sep 2015

Quadratic invariants of the elasticity tensor Yakov Itin Inst. Mathematics, Hebrew Univ. of Jerusalem, Givat Ram, Jerusalem 91904, Israel, and Jerusalem College of Technology, Jerusalem 91160, Israel, email: [email protected] September 9, 2015, file Quadratic6.tex

Abstract We study the quadratic invariants of the elasticity tensor in the framework of its unique irreducible decomposition. The key point is that this decomposition generates the direct sum reduction of the elasticity tensor space. The corresponding subspaces are completely independent and even orthogonal relative to the Euclidean (Frobenius) scalar product. We construct a basis set of seven quadratic invariants that emerge in a natural and systematic way. Moreover, the completeness of this basis and the independence of the basis tensors follow immediately from the direct sum representation of the elasticity tensor space. We define the Cauchy factor of an anisotropic material as a dimensionless measure of a closeness to a pure Cauchy material and a similar isotropic factor is as a measure for a closeness of an anisotropic material to its isotropic prototype. For cubic crystals, these factors are explicitly displayed and cubic crystal average of an arbitrary elastic material is derived.

Key index words: anisotropic elasticity tensor, irreducible decomposition, quadratic invariants

1

Introduction

In the linear elasticity theory of anisotropic materials, the relation between the strain tensor uij and the stress tensor σ ij , the generalized Hooke’s law, is expressed by the use of the elasticity (stiffness) tensor C ijkl σ ij = C ijkl ukl . (1) In 3-dimensional space, a generic 4th order tensor has 81 independent components. However, due to the standard symmetry assumptions for the stress and strain tensors, C ijkl = C jijkl = C klij ,

(2)

the elasticity tensor is left with 21 independent components only. These components are not really the intrinsic characteristics of the material because they depend on the choice of the coordinate system. Thus, in order to deal with the proper material parameters, one must look for the invariants of the elasticity tensor.

1

Y. Itin

Quadratic invariants of the elasticity tensor

2

There are only two linearly independent invariants of the first order of C ijkl . They are usually taken as follows, see [20], ∗

A1 = C ij ij = Cijij

∗

A2 = Ci ij j = Ciijj .

and

(3)

Here and subsequently, we use the standard tensor conventions that strictly distinguish between covariant and contravariant indices. In these notations, two repeated indices can appear only in updown positions and summation for only two such repeated indices is assumed. The indices of a tensor can be raised/lowered by the use of the metric tensor g ij /gkl . For instance, the lower components 0 0 0 0 of the elasticity tensor are defined as Cijkl = gii0 gjj 0 gkk0 gll0 C i j k l . Since in the elasticity literature a simplified notation is frequently used, we provide both notations in most cases and relate the ∗ corresponding quantities by the sign = , as in (2). Notice that this shorthand notation is applicable only in Euclidean space endowed with rectangular coordinates. Quadratic invariants of the elasticity tensor were studied by Ting [25]. He presented two such invariants, ∗

B1 = C ijkl Cijkl = Cijkl Cijkl

∗

B2 = Ci ikl C j jkl = Ciikl Cjjkl .

and

(4)

Ahmad [1] has contributed the two additional quadratic invariants, ∗

B3 = Ci ikl C j kjl = Ciikl Cjkjl

∗

B4 = Ci kli C j klj = Ckiil Ckjjl .

and

He also proved that the set of seven quadratic invariants A21 , A22 , A1 A2 , B1 , B2 ,

(5)

B3 ,

B4

(6)

is linearly independent. Norris [20] studied the problem of the quadratic invariants and proved that the set (6) is complete. It means that every quadratic invariant of the elasticity tensor is a linear combination of the seven invariants listed in (6). In order to prove this fact, Norris presented a generic quadratic invariant in the form I = fijklpqrs C ijkl C pqrs ,

(7)

where fijklpqrs is a numerical tensor and provided a detailed analysis of this tensor. It is proven that due to the symmetries (2), the most components of fijklpqrs vanish and a lot of the remaining components are linearly related. This way, exactly seven invariants are left over. In particular, Norris demonstrated that a rather natural additional invariant ∗

B5 = C ijkl Cikjl = Cijkl Cikjl

(8)

can be in fact represented as a linear combination of the invariants listed in (6), namely 1 1 B5 = A21 + A22 − A1 A2 + B1 − 2B2 + 4B3 − 2B4 . 2 2 In this situation, some principal questions arise: • Is there some preferable basis of quadratic invariants? • Is there a systematic way to construct such a basis?

(9)

Y. Itin


3

• Which quadratic invariants can be used as characteristic parameters for different elastic materials? • Which physical interpretation can be given to various quadratic invariants of the elasticity tensor? In the current paper, we analyze the quadratic invariants problem in the framework of the unique irreducible decomposition of the elasticity tensor. In Section 2, we present the principle algebraic facts of this decomposition. The key point that such resolution of the elasticity tensor in the simple pieces generates the direct sum reduction of the elasticity tensor space. The corresponding subspaces are completely independent and even orthogonal relative to the Euclidean (Frobenius) scalar product. In this framework, we construct in Section 3 the basis set of seven quadratic invariants that emerge in a natural and systematic way. Moreover the completeness of this basis and its independence of basis tensors follow immediately from the direct sum representation of the elasticity tensor space. We compare this basis to the basis given in (6). In section 4, we provide some applications of quadratic invariants to physics motivated problems. We define Cauchy factor for an arbitrary anisotropic material. It can be used as a measure of deviation of a material from an analogical Cauchy material. We also prove Fedorov’s relation for an isotropic material closest to an anisotropic one. This result follows immediately from the irreducible orthogonal decomposition. Correspondingly, we define the isotropic factor that measures the closeness of an anisotropic material to its isotropic prototype. In section 5, we study a simplest non-trivial example of cubic crystal. It is naturally represented by three independent quadratic invariants. This low-dimensional case allows the visualization of the Cauchy and isotropic factors. Corresponding graphs are presented. In Conclusion section we present our main results and propose some possible direction for future investigations.

2

Irreducible decomposition

To describe the irreducible decomposition of the elasticity tensor we first observe two groups acting on it simultaneously, see [15]. For an arbitrary tensor of the range p defined on Rn , these are the permutation (symmetry) group Sp and the group of rotations SO(n, R).

2.1

Irreducible decomposition under the permutation group

The symmetry group S4 provides permutations of the elasticity tensor indices. The decomposition of C ijkl under this group is described by two Young diagrams: ⊕

⊗

⊗

=

⊕

.

(10)

All other 4-the order Young’s diagrams are non-relevant in our case due to the original symmetries (2) of C ijkl . The left-hand side of (10) represents a generic 4th rank tensor. On the right-hand side, two diagrams describe two tensors of different symmetries. The explicit expression of these tensors can be computed by the use of the corresponding symmetrization and antisymmetrization operations. For the elasticity tensor, the result is obvious. The first (row) diagram represents the totally symmetric tensor. The second (square) diagram represents an additional tensor that can be considered merely as a remainder.

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4

As a result, we arrive at the unique and irreducible decomposition of the elasticity tensor under the action of the group S4 : C ijkl = S ijkl + Aijkl , (11) with

1 ijkl C + C iklj + C iljk 3

S ijkl := C (ijkl) =

(12)

and

1 ijkl 2C − C ilkj − C iklj . (13) 3 We use the standard normalized Bach parentheses for symmetrization and antisymmetrization of indices. We observe that every antisymmetrization of S ijkl gives zero and every symmetrization preserves this tensor. As for the second part Aijkl , its total symmetrization A(ijkl) vanishes. Consequently, we have a useful symmetry relation Aijkl := C ijkl − C (ijkl) =

Aijkl + Aiklj + Ailjk = 0 .

(14)

As it is shown in [13] and [14], the equation Aijkl = 0 describes the well-known Cauchy relation. Thus, we call S ijkl the Cauchy part and Aijkl the non-Cauchy part of the elasticity tensor . Observe the main algebraic properties of this decomposition: • The partial tensors S ijkl and Aijkl satisfy the minor symmetries, S [ij]kl = S ij[kl] = 0

and

A[ij]kl = Aij[kl] = 0 ,

(15)

and

Aijkl = Aklij .

(16)

and the major symmetry, S ijkl = S klij

Thus, these partial tensors can themselves serve as elasticities of some hypothetic material. • Moreover, any additional symmetrization or antisymmetrization preserves the tensors S ijkl and Aijkl or nullifies them. • The decomposition (11) is preserved under arbitrary linear transformations. Thus, it can be referred to as irreducible GL(3, R)-decomposition. • The irreducible decomposition of the tensor C provides the decomposition of the corresponding tensor space C into a direct sum of two subspaces S ⊂ C (for the tensor S) and A ⊂ C (for the tensor A), C = S ⊕ A. (17) In particular, we have dim C = 21 ,

dim S = 15 ,

dim A = 6 .

(18)

• The irreducible pieces S ijkl and Aijkl are orthogonal to one another in the following sense: S ijkl Aijkl = 0 .

(19)

S ijkl Aijkl = S (ijkl) Aijkl = S (ijkl) A(ijkl) = 0 .

(20)

Indeed,

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5

• “Pythagorean theorem:” The Euclidean (Frobenius) squares of the tensors e 2 = C ijkl Cijkl , C

Se2 = S ijkl Sijkl ,

e2 = Aijkl Aijkl A

(21)

satisfy the relation e 2 = Se2 + A e2 . C

2.2

(22)

Irreducible decomposition under the rotation group

We are looking now for the decomposition of the elasticity tensor under the action of the group of rotations. The following fact is due to the classical theory of invariants: Relative to the subgroup SO(3) of GL(3), a basis of an arbitrary system of tensors coincides with a basis of the same system with the metric tensor added [27]. We use the Euclidean metric tensor gij . In rectangular coordinates, gij = diag(1, 1, 1). We start with the totally symmetric Cauchy part S ijkl . From the contraction of S ijkl with the metric tensor, we construct a unique symmetric second-rank tensor ∗

S ij := gkl S ijkl = Sijkk =

1 (Cijkk + 2Cikkj ) 3

(23)

and a unique scalar ∗

S := gij S ij = Siikk =

1 (Ciikk + 2Cikki ) . 3

(24)

We denote the traceless part of the tensor S ij as 1 P ij := S ij − Sg ij , 3

gij P ij = 0 .

with

(25)

Now we turn to the decomposition of the tensor S ijkl . We denote the two subtensors (1) ijkl

S

:= αSg (ij g kl) ,

(2) ijkl

S

:= βP (ij g kl) .

(26)

It can be checked now by the straightforward calculations that the remainder Rijkl := S ijkl − (1)S ijkl − (2)S ijkl

(27)

is totally traceless if and only if α=

1 , 5

β=

6 . 7

(28)

Hence, we obtain the decomposition of the totally symmetric tensor S ijkl into the sum of three independent pieces: S ijkl = (1)S ijkl + (2)S ijkl + (3)S ijkl , (29) where (1) ijkl

S

(2) ijkl

S

6 = P (ij g kl) 7

1 1 = Sg (ij g kl) = S g ij g kl + g ik g jl + g il g jk . 5 15 1 = P ij g kl + P ik g jl + P il g jk + P jk g il + P jl g ik + P kl g ij . 7

(30) (31)

and (3) ijkl

S

= Rijkl .

(32)

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6

These pieces are unique and invariant under the action of the group SO(3). Moreover, the corresponding subspaces (1)S, (2)S and (3)S are mutually orthogonal. Indeed, for A, B = 1, 2, 3 with A 6= B, (A) Sijkl (B)S ijkl = 0 . (33) This fact follows immediately from the tracelessness of the tensors P ij and Rijkl . Consequently, the vector space S of the totally symmetric tensor S ijkl is decomposed into the direct sum of three subspaces S = (1)S ⊕ (2)S ⊕ (3)S (34) with the corresponding dimensions 15 = 1 ⊕ 5 ⊕ 9 .

(35)

We turn now to the second part of the elasticity tensor. The irreducible piece Aijkl is a fourth rank tensor with 6 independent components. It is quite naturally that it can be represented as a symmetric second-rank tensor, see [3],[12], [13], and [14] for detailed discussions. We define 1 ∆mn := mil njk Aijkl , 3

(36)

where ijk = 0, ±1 denotes the 3-dimensional Levi-Civita permutation pseudo-tensor. Consequently, ∆mn is a symmetric tensor, ∆mn = ∆nm , and we have Aijkl = im(k l)jn ∆mn .

(37)

For a proof of this proposition it is enough to substitute (36) into (37) and to apply the standard relations for the Levi-Civita pseudo-tensor. In order to decompose the non-Cauchy part Aijkl , it is convenient to use its representation by the tensor density ∆ij . We denote A := g mn ∆mn . (38) By using the relation g mn mil njk = gij glk − gik gjl

(39)

and Eq.(37), we derive A=

1 1 ∗ 1 (gij glk − gik gjl ) Aijkl = (Aiikk − Aikik ) = (Ciikk − Cikik ) . 3 3 3

(40)

The tensor density ∆ij can be decomposed into the scalar and traceless pieces: 1 ∆ij = Qij + Agij , 3

(41)

1 Qij := ∆ij − Agij . 3

(42)

Aijkl = (1)Aijkl + (2)Aijkl ,

(43)

where the traceless piece is given by

Substituting (41) into (37), we obtain

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7

where the scalar part is given by (1) ijkl

A

1 := A 2g ij g kl − g il g jk − g ik g jl 6

(44)

and the remainder reads

1 ikm jln + ilm jkn Qmn . (45) 2 We recall that the tensor Qmn is symmetric and traceless. Since the product of two Levi-Civita pseudo-tensors is represented by the determinant of the metric tensor g ij , the latter equation can be rewritten as (2) ijkl

A

(2) ijkl

A

:=

:= g ik Qjl + g jk Qil + g il Qjk + g jl Qik − 2g kl Qij − 2g ij Qkl .

(46)

The decomposition given in Eq.(43) is unique, invariant, and irreducible under the action of the rotation group SO(3, R) and of the permutation group S4 . Correspondingly, the vector space A of the tensor Aijkl is irreducibly decomposed into the direct sum of two subspaces A = (1)A ⊕ (2)A , (47) with the corresponding dimensions 6 = 1 ⊕ 5. Since the trace of

(2)Aijkl

(48)

equals zero, these subspace are orthogonal to one another, (1)

Aijkl (2)Aijkl = 0 .

(49)

Collecting our results, we formulate the following Theorem 1. Under the simultaneous action of the groups S4 and SO(3, R), the elasticity tensor is uniquely irreducibly decomposed into the sum of five parts C

ijkl

=

5 X

(A)

C ijkl =

(1) ijkl

S

+ (2)S ijkl + (3)S ijkl + (1)Aijkl + (2)Aijkl .

(50)

A=1

This decomposition corresponds to the direct sum decomposition of the vector space of the elasticity tensor into five subspaces C = (1) C ⊕ (2) C ⊕ (3) C ⊕ (4) C ⊕ (5) C , (51) with the dimensions 21 = (1 ⊕ 5 ⊕ 9) ⊕ (1 ⊕ 5) ,

(52)

The irreducible pieces are orthogonal to one another: For A 6= B (A)

Cijkl (B) C ijkl = 0 .

(53)

The Euclidean squares, C 2 = Cijkl C ijkl and (A) C 2 = (A) Cijkl (A) C ijkl with A = 1, · · · , 5, fulfill the “Pythagorean theorem:” C 2 = (1) C 2 + (2) C 2 + (3) C 2 + (4) C 2 + (5) C 2 . (54)

Y. Itin

2.3


8

Irreducible decompositions

The decomposition (51) involves two scalars S and A, two second order traceless tensors Pij and Qij , and a fourth order totally traceless tensor Rijkl . Exactly the same types of tensors emerge in the harmonic decomposition that is widely used in elasticity theory. Such decomposition is generated by expressing the partial tensors in term of the harmonic polynomials, i.e., the polynomial solutions of the Laplace equation. The corresponding tensors are required to be completely symmetric and totally traceless. As it was demonstrated by Backus [3], such a harmonic decomposition is not applicable in general in a space of the dimension greater than three. The most compact expression of this type was proposed by Cowin [8], C ijkl = ag ij g kl + b g ik g jl + g il g jk + g ij Aˆkl + g kl Aîj + ik ˆ jl jk ˆ il il ˆ jk ik ˆ jl + Z ijkl . (55) g B +g B +g B +g B An alternative expression was proposed by Backus [3]. It reads, see [4], C ijkl = H ijkl + H ij g kl + H ik g jl + H il g jk + H jk g il + H jl g ik + H kl g ij + H g ij g kl + g ik g jl + g il g jk + 1 jl ik 1 ik jl 1 jk il 1 il jk ij kl kl ij h g +h g − h g − h g − h g − h g + 2 2 2 2 1 il jk 1 ik jl ij kl h g g − g g − g g 2 2

(56)

Let us compare these two expressions. First we observe that two totally traceless tensors must be equal to one another, H ijkl = Z ijkl . As for the scalar terms in Eq.(55), they are merely linear combinations of the corresponding terms in Eq.(56). Indeed, it is enough to take a = H + h and b = H − (1/2)h .

(57)

Quite similarly, the traceless second order tensors of Eq.(55) are linear combinations of the corresponding terms of Eq.(56) with the identities Aˆkl = H kl + hkl

ˆ jl = H jl − (1/2)hjl . and B

(58)

Both decompositions are irreducible under the action of the rotation group. The key difference between the two is that that the decomposition of Backus is also irreducible under the action of permutation group. Cowin’s decomposition is reducible in this sense. Let us compare now the harmonic decomposition Eq.(56) to our decomposition as it is given in Eq.(50). We immediately identify (1) ijkl S = H g ij g kl + g ik g jl + g il g jk , (59) (2) ijkl

S

(3) ijkl

S

(1) ijkl

A

(2) ijkl

A

= H ij g kl + H ik g jl + H il g jk + H jk g il + H jl g ik + H kl g ij , ijkl

= H , 1 il jk 1 ik jl ij kl = h g g − g g − g g , 2 2 1 1 1 1 = hij g kl + hkl g ij − hjl g ik − hik g jl − hjk g il − hil g jk . 2 2 2 2

(60) (61) (62) (63)

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9

We can straightforwardly derive the relations H=

1 S, 15

1 H ij = P ij , 7

H ijkl = Rijkl ,

(64)

and

1 h = A, hij = −2Qij . (65) 3 Thus the two decompositions are equivalent. The difference is that in Eq.(50) the partial tensors are identified as elasticities themselves. These partial tensors generate the minimal direct sum decomposition of the elasticity tensor space. Moreover. the corresponding subspaces are mutually orthogonal and the squares of the tensors satisfy the “Pythagorean theorem”. We will see in the following section how those properties can be applied to the problem of quadratic invariants.

3 3.1

Linear and quadratic invariants Linear invariants

Each linear invariant of the elasticity tensor can be represented as I = hijkl C ijkl ,

(66)

where hijkl is a tensor. Using the decomposition (51), it can be rewritten as a sum of irreducible parts with different leading coefficients I=

5 X

(I)

hijkl (I) C ijkl .

(67)

I=1

We observe that the tensor hijkl can be constructed only as a product of two components of the metric tensor, hijkl ∼ (g · g)ijkl . Since the tensors Pij , Qij , and Rijkl are totally traceless they do not contribute to the sum in Eq.(67). Consequently we are left with I = αS + βA .

(68)

Hence, every linear invariant is represented as a linear combination of the two basic linear invariants ∗

L1 = S =

1 (Ciikk + 2Cikik ) , 3

∗

L2 = A =

1 (Ciikk − Cikik ) . 3

(69)

Their independence can be seen from their explicit expressions. It follows also from the fact that S and A are related to two different subspaces (1) S and (1) A. We readily obtain the expression of the linear invariants given in Eq.(2), Cikik = S − A ,

Ciikk = S + 2A .

(70)

Y. Itin

3.2


10

Quadratic invariants

Norris [20] presented a generic quadratic invariant of the elasticity tensor in the form fijklpqrs C ijkl C pqrs .

(71)

Here fijklpqrs is a numerical tensor. Using the irreducible decomposition (51) it can be rewritten as 5 X

(I,J)

fijklpqrs (I)C ijkl

(J) pqrs

C

.

(72)

I,J=1

The tensors with the components (I,J)fijklpqrs can be constructed only as a product of four components of the metric tensor, (I,J)fijklpqrs ∼ (g · g · g · g)ijklpqrs . Using the traceless property of the tensors Pij , Qij , and Rijkl , we can show that the quadratic invariants can be chosen uniquely as Z1 = S 2 , Z4 = Pij P ij ,

Z2 = AS ,

Z5 = Pij Qij ,

Z3 = A2 ,

Z6 = Qij Qij ,

and

(73) Z7 = Rijkl Rijkl .

(74)

It is clear that this set of invariants is complete. Indeed, when two tensors in (72) are irreducibly decomposed in the form (51) with an arbitrary tensor fijklpqrs constructed from the metric tensor and numbers, only the terms (73, 74) can appear. Moreover, these invariants are independent. It is due to the fact that they are taken from independent and even orthogonal subspaces of the elasticity tensor space.

3.3

Relations between two sets of quadratic invariants

Let us display the quadratic invariants of the set (6) in terms of ZI . We present the details of calculations in the Appendix. For the quadratic invariants constructed from the linear ones, we have straightforwardly A21 = (S − A)2 = Z1 − 2Z2 + Z3 ,

(75)

A1 A2 = (S − A)(S + 2A) = Z1 + Z2 − 2Z3 , A22

2

= (S + 2A) = Z1 + 4Z2 + 4Z3 .

(76) (77)

For the first invariant of Ting, we write B1 = C ijkl Cijkl =

5 X

(I) ijkl

C

5 X

(J)

Cijkl .

(78)

B1 = (1) S 2 + (2) S 2 + (3) S 2 + (4) A2 + (5) A2

(79)

I=1

J=1

Due to the orthogonality of the set, we are left here with

Let us list the expressions of these invariants 1 S = S2 , 5

(1) 2

6 S = P ij Pij , 7

(2) 2

(3) 2

S = Rijkl Rijkl ,

(80)

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11

and (1)

A2 = A2 ,

(2)

A2 = 12Qmn Qmn .

(81)

Consequently, 1 6 B1 = Z1 + Z3 + Z4 + 12Z6 + Z7 . 5 7 The second invariant of Ting takes the form 1 4 4 B2 = Z1 + Z2 + Z3 + Z4 − 4Z5 + 4Z6 . 3 3 3

(82)

(83)

The first and the second invariants of Ahmad read 1 2 1 B3 = Z1 + Z2 − Z3 + Z4 − Z5 − 2Z6 3 3 3

(84)

and

1 2 1 B4 = Z1 − Z2 + Z3 + Z4 + 2Z5 + Z6 , (85) 3 3 3 respectively. From these expressions we see that the set of invariants (6) is complete and the invariants are independent. The same is true for our set Zi . We calculate also the additional invariant of Norris: 1 1 6 B5 = C ijkl Cikjl = Z1 − Z3 + Z4 − 6Z6 + Z7 . (86) 5 2 7 Substituting the expressions (73–74) we obtain the formula (9) of Norris.

4

Applications: Invariants as characteristics of materials

Since invariants of the elasticity tensor are independent of the coordinate system used in specific measurements, they can be used as intrinsic characteristics of the materials. It is clear that linear independence is not enough for this goal. Indeed, although the invariants Z1 , Z2 , and Z3 are linear independent, they are related by a quadratic relation Z22 = Z1 Z3 . We will show that an intrinsic meaning can be assigned to the five invariants that correspond to different direct subspaces of the elasticity tensor space, namely {Z1 , Z3 , Z4 , Z6 , Z7 } . (87) All these invariants are positive.

4.1

Cauchy relations and Cauchy factor

In the early days of the elasticity theory, Cauchy formulated a molecular model for elastic bodies, based on 15 independent elasticity constants. In this way 6 constraints, called Cauchy relations were assumed. A lattice-theoretical analysis shows, see [12], [16] , that the Cauchy relations are valid provided the following conditions hold: • The interaction forces between the molecules of a crystal are central forces; • each molecule is a center of symmetry; • the interaction forces between the building blocks of a crystal can be well approximated by a harmonic potential.

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12

More recent discussions of the Cauchy relations can be found, e.g., in [1], [3], [4], or [7]. Different compact expressions of the Cauchy relations can be found in literature. For instance in [12], they are presented as C iijk − C ijik = 0 . (88) An alternative form is widely used, see [22], [26], [10], [7], C ijkl − C ikjl = 0 .

(89)

The irreducible decomposition technique [13] yields Aijkl = 0 ,

or

Qij = 0 .

(90)

As it was demonstrated experimentally already by Voigt, the Cauchy relations do not hold even approximately. Thus, elastic properties of the generic anisotropic material is described by the whole set of 21 independent component. In fact, the situation with the Cauchy relations is much more interesting, see [12]. One can look for the deviation of the elasticity tensor from its Cauchy part. As it was pointed out by Hauss¨ uhl [12], this deviation, even being a macroscopic characteristic, can provide some important information about microscopic structure of the material. To have such deviation term we must have a unique proper decomposition of the elasticity tensor into two independent parts that can be referred to as Cauchy and non-Cauchy parts. In [12], the deviation from the Cauchy part was presented by the value of the corresponding combination given in the left hand side of Eq.(88). Such way of expression is valid only in the case when the non-Cauchy part is presented by only one component. Moreover this expression is dimension-full and depends on the choice of the coordinate system. With the use of the quadratic invariants we can introduce an invariant characteristic of deviation of a material from its Cauchy prototype. Due to the “Pythagorean theorem” (51), we define the dimensionless quantity, which we will call the Cauchy factor s S ijkl Sijkl FCauchy = . (91) C ijkl Cijkl Evidently, 0 ≤ FCauchy ≤ 1. A pure Cauchy material is determined by FCauchy = 1. For FCauchy = 0, we have a hypothetic material without Cauchy part at all. Comparing two materials, we must conclude that a material with higher Cauchy factor has a microscopic structure closer to spherical symmetry.

4.2

Fedorov’s problem

In linear elasticity for anisotropic materials one must deal with a big set of elasticity constants. But in some problems, the elastic body can only be slightly different from an isotropic one. Fedorov [11], in a classical book on the propagation of elastic waves in anisotropic crystals, has demonstrated how the anisotropic elastic tensor can be averaged over the 3-dimensional spatial directions in order to find some kind of isotropic approximation. Recently Norris [21] took up this program and defined an Euclidean distance function for solving the Fedorov problem in a novel way. He succeeded in doing so and even extended the formalism for averaging the given set of elastic parameters relative to less symmetric classes. The corresponding procedure can be outlined as follows:

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13

• For a given 4th order elasticity tensor, one constructs the corresponding 2nd order Christoffel tensor, which is quadratic in the wave vector. • One consider the C 2 norm of the difference between the given anisotropic Christoffel tensor and a generic isotropic one. The isotropic Christoffel tensor is taken with two unknown parameters. • Since Christoffel tensor is quadratic in the wave vector n, the mentioned norm is quartic n likewise. Its average is computed in space directions. • The resulting expression is left to be a function of two isotropic parameters. Its minimization is applied and the resulting pair of isotropic parameters is derived. The result of this consideration is given in [21] as ∗

κ=

1 Ciijj , 9

∗

µ=

1 1 Cijij − Ciijj . 10 30

(92)

In [20], [21] and [18], Norris explained that minimizing a Euclidean distance function is equivalent to projecting the tensor of elastic stiffness onto the appropriate symmetry. Let us consider an elasticity tensor of 21 independent components. It is irreducibly decomposed to the sum of five independent pieces. Two scalar pieces, namely (1)S ijkl and (1)Aijkl has a special property: They are invariant under arbitrary SO(3) transformation. Using the direct sum of the corresponding subspaces, we can construct a subspace ISO = (1) S ⊗ (1) A .

(93)

This 2-dimensional subspace is SO(3) invariant and orthogonal to all other subspaces of an arbitrary elasticity tensor. Evidently it must be identified as an isotropic part of an elasticity tensor. Consequently we constructed an isotropic part of the generic elasticity tensor in the form (iso)

C ijkl = (1)S ijkl + (1)Aijkl ,

(94)

or, explicitly, (iso)

C ijkl =

1 1 ij kl S g g + g ik g jl + g il g jk + A 2g ij g kl − g il g jk − g ik g jl . 15 6

(95)

Let us compare this expression to the standard representation of an isotropic material in terms of the Lamé moduli λ and µ (iso) ijkl C = λg ij g kl + µ g ik g jl + g il g jk . (96) We derive the effective Lame moduli for an anisotropic material λ= Recall that ∗

S=

1 1 S + A, 15 3

1 (Ciikk + 2Cikik ) , 3

µ=

∗

A=

1 1 S − A. 15 6 1 (Ciikk − Cikik ) . 3

(97)

(98)

Y. Itin


14

Substituting these expressions into equations (97), we derive ∗

λ=

2 1 Ciijj − Cijij , 15 15

∗

µ=

1 1 Cijij − Ciijj . 10 30

(99)

With the bulk constant κ = λ + (2/3)µ we recover both expressions given in Eq.(92). In order to express the deviation of the given anisotropic material from its effective isotropic prototype, one uses the distance between two tensors. This quantity is dimensionful and depends on the average magnitude of the elasticity tensor. Instead, we define the isotropy factor of an anisotropic material in the form s s (1) S (1) ijkl + (1) A (1) ijkl (iso) C 2 ijkl S ijkl A Fiso = = . (100) ijkl ijkl Cijkl C Cijkl C In terms of the constants S and A it reads s s S 2 + 5A2 Z1 + 5Z3 = Fiso = . ijkl Z1 + 5Z3 + (30/7)Z4 + 60Z6 + 5Z7 5Cijkl C

(101)

We observe that 0 ≤ Fiso ≤ 1. It is equal to one for pure isotropic materials and equal to zero for some hypothetic material without isotropic part, i.e., in the case when the effective Lamé moduli vanish.

4.3

Irreducibility factors

As a natural extension of the Cauchy and the isotropy factors described above, we introduce dimensionless numerical factors that describe the contribution of the irreducible pieces to the elasticity tensor. For the 5 irreducible parts (I) C ijkl with I = 1, · · · , 5, we define the irreducibility factors s (I) C ijkl (I) C ijkl (I) Firr = . (102) Cijkl C ijkl In particular, the Cauchy factor is expressed as s q (2) ijkl (1) ijkl + (2) A (1) A ijkl A ijkl A (4) F 2 + (5) F 2 FCauchy = = irr irr Cijkl C ijkl and the isotropy factor as s Fiso =

(1) S (1) ijkl ijkl S

+ (1) Aijkl (1) Aijkl = Cijkl C ijkl

q

(1) F 2 irr

2 . + (4) Firr

(103)

(104)

Y. Itin

5 5.1


15

Cubic crystals Definition

Cubic crystals are described by three independent elasticity constants. In a properly chosen coordinate system, they can be put, see Nayfeh [19], into the following Voigt matrix:  1111   11  C C 1122 C 1133 C 1123 C 1131 C 1112 C C 12 C 12 0 0 0 11 C 12   ∗ C 2222 C 2233 C 2223 C 2231 C 2212  0 0 0     ∗ C  3333 3323 3331 3312 11    ∗ ∗ C C C C ∗ C 0 0 0   ≡ ∗  . (105)   ∗ ∗ ∗ C 2323 C 2331 C 2312  ∗ ∗ C 66 0 0     ∗   ∗ ∗ ∗ ∗ C 3131 C 3112   ∗ ∗ ∗ ∗ C 66 0  ∗ ∗ ∗ ∗ ∗ C 1212 ∗ ∗ ∗ ∗ ∗ C 66 Taking into account the multiplicities of the elements exhibited in (105), we calculate 2 2 2 C 2 = Cijkl C ijkl = 3 C 11 + 6 C 12 + 12 C 66 .

5.2

(106)

S4 -decomposition

We decompose (105) irreducibly and  1111 S S 1122 S 1133  ∗ S 2222 S 2233   ∗ ∗ S 3333 S ijkl =   ∗ ∗ ∗   ∗ ∗ ∗ ∗ ∗ ∗

find the Cauchy part   α S 1123 S 1131 S 1112 2223 2231 2212   S S S  ∗  S 3323 S 3331 S 3312   = ∗ 2323 2331 2312   S S S  ∗ 3131 3112  ∗ ∗ S S 1212 ∗ ∗ ∗ S

where α = C 11 ,

β=

β α ∗ ∗ ∗ ∗

β β α ∗ ∗ ∗

 0 0 0 0 0 0  0 0 0 , β 0 0  ∗ β 0 ∗ ∗ β

1 C 12 + 2C 66 , 3

(107)

(108)

The square of this tensor takes the value Se2 = Sijkl S ijkl = 3α2 + 18β 2 = 3 C 11 The non-Cauchy part is represented by  1111 A A1122 A1133 A1123  ∗ A2222 A2233 A2223   ∗ ∗ A3333 A3323 Aijkl =   ∗ ∗ ∗ A2323   ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

A1131 A2231 A3331 A2331 A3131 ∗

where γ=

2

+ 2 C 12 + 2C 66

2

.

   A1112 0 2γ 2γ 0 0 0  0 0  A2212    ∗ 0 2γ 0 3312    ∗ ∗ 0 0 0 0 A , = 2312   0  A   ∗ ∗ ∗ −γ 0 ∗ −γ 0  A3112  ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −γ A1212

1 C 12 − C 66 . 3

(109)

(110)

(111)

Its square reads e2 = Aijkl Aijkl = 36γ 2 = 4 C 12 − C 66 A

2

.

(112)

Y. Itin


16

We can straightforwardly check the identities e · Se = 0 , A

(113)

e2 = 3α2 + 18β 2 + 36γ 2 . C 2 = Se2 + A

(114)

and The Cauchy factor is expressed as s FCauchy = or

s FCauchy =

α2 + 6β 2 , α2 + 6β 2 + 12γ 2

3 (C 11 )2 + 2 (C 12 + 2C 66 )2 3 (C 11 )2 + 6 (C 12 )2 + 12 (C 66 )2

(115)

.

(116)

For the Cauchy relation we have γ = 0,

or

C 12 = C 66 .

(117)

In this case, Eq.(115) yields FCauchy = 1. For cubic crystals, the Cauchy factor in Eqs.(115,116) depends on two independent parameters and thus allows 3-dimensional visualization, see Fig. 1 and Fig 2.

Figure 1: Functional dependence of the Cauchy factor FCauchy with respect to the parameters β/α and γ/α. The pure Cauchy materials are depicted by the straight line lying on the axis γ = 0.

5.3

SO(3)-decomposition

Let us determine the SO(3) components of the totally symmetric tensor. Using Eq.(23) we derive from (107) 2 12 4 66 ij ij ij 11 g . (118) S = (α + 2β)g = C + C + C 3 3 Consequently, S = 3(α + 2β) = 3C 11 + 2C 12 + 4C 66 ,

(119)

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17

Figure 2: Functional dependence of the Cauchy factor FCauchy with respect to the parameters C 12 /C 11 and C 66 /C 11 . The pure Cauchy materials are depicted by the straight line C 12 = C 66 . and P ij = 0 . The total traceless remainder can be expressed  1111 R R1122 R1133 R1123 R1131  ∗ R2222 R2233 R2223 R2231   ∗ ∗ R3333 R3323 R3331 Rijkl =   ∗ ∗ ∗ R2323 R2331   ∗ ∗ ∗ ∗ R3131 ∗ ∗ ∗ ∗ ∗

(120)

as  R1112 R2212   R3312   = α − 3β 2312  R 5  3112  R R1212

  2 −1 −1 0 0 0 ∗ 2 −1 0 0 0   ∗ ∗  2 0 0 0  , ∗ ∗ ∗ −1 0 0   ∗ ∗ ∗ ∗ −1 0  ∗ ∗ ∗ ∗ ∗ −1 (121)

The two non-zero quadratic invariants read 2

(122)

2 6 6 C 11 − C 12 − C 66 . R2 = (α − 3β)2 = 5 5

(123)

S 2 = 9(α + 2β)2 = 3C 11 + 2C 12 + 4C 66 and

We check the formula

1 S˜2 = S 2 + R2 , (124) 5 that must hold with the correspondence with the expressions (80,81) of the first Ting’s invariant. Let us turn now to the asymmetric part Aijkl . Eq.(110) yields ∆ij = 2γg ij .

(125)

Consequently, A = 6γ ,

Qij = 0 .

(126)

Thus, e2 = A2 = 36γ 2 = 4 C 12 − C 66 A

2

.

(127)

Y. Itin


Accordingly, the isotropic factor takes the form s s (1/5)S 2 + A2 (3/5)(α + 2β)2 + 12γ 2 , = Fiso = α2 + 6β 2 + 12γ 2 Cijkl C ijkl or Fiso

v u u (3C 11 + 2C 12 + 4C 66 )2 + 20 (C 12 − C 66 )2 =t . 15 (C 11 )2 + 2 (C 12 )2 + 4 (C 66 )2

18

(128)

(129)

It is well known that an isotropic medium can be described as a cubic crystal with C 66 = (1/2)(C 11 − C 12 ). It is equivalent to α = 3β. When this relation is substituted into (128), we obtain Fiso = 1. In Fig. 3 and Fig 4. we present the functional dependence of the isotropy factor for a cubic crystal with respect to the homogeneous fractions of its parameters.

Figure 3: Functional dependence of the isotropy factor Fiso is depicted with respect to the variables β/α and γ/α. The straight line with β/α = 1/3 corresponds to the pure isotropic materials.

Figure 4: Functional dependence of the isotropy factor Fiso with respect to the variables C 12 /C 11 and C 66 /C 11 . The straight line C 66 = (1/2)(C 11 − C 12 ) correspond to the isotropic medium.

Y. Itin

5.4


19

Cubic averaging

We consider now a problem of averaging of an arbitrary elastic material by a cubic symmetry prototype. Let in some coordinate system a material be described by a full set of 21 elastic parameters C ijkl . We are looking for a cubic crystal that is mostly close to our material with respect to the Euclidean metric. We assume that in the same coordinate system the cubic crystal is represented in the canonical form (105). Since for cubic crystal P ij = Qij = 0, its elasticity tensor is left ijkl ijkl ijkl Ccub = (1) Scub + (3) Scub + (1) Aijkl cub ,

(130)

where (1) ijkl Scub (1) ijkl Acub (3) ijkl Scub

1 ij kl m g g + g ik g jl + g il g jk , 15 1 ij kl n 2g g − g il g jk − g ik g jl , = 6 = kσ ijkl . =

(131) (132) (133)

Here σ ijkl is a set of parameters (not a tensor) that is represented in the 6 × 6 notations by the constant matrix in the right hand side of Eq.(121). Thus m, k, n are our unknown variables. The square distance between two elasticity tensors is given by

ijkl C ijkl − Ccub

2

= K1 (S − m)2 + K2 (A − n)2 + (Rijkl − kσ ijkl )2 + K3 ,

(134)

where K1 , K2 , K3 are positive numerical constants. This expression reaches its minimal value for m=S,

n = A,

(135)

and (Rijkl − kσ ijkl )σijkl = 0 , We have k=

Rijkl σijkl , σ ijkl σijkl

(136)

(137)

Using the expression of σ ijkl listed in Eq.(121), we derive k=

R1111 + R2222 + R3333 . 6

(138)

Consequently the parameters of the cubic crystal prototype take the values α = β = γ =

1 1 2 S + 2k = C 1111 + C 2222 + C 3333 − S 5 3 15 1 1 7 1111 2222 3333 S−k =− C +C +C − S 15 6 30 1 A. 6

(139) (140) (141)

Y. Itin

6


20

Conclusion

In the framework of the irreducible decomposition of elasticity tensor, we studied the problem of its quadratic invariants. Since this decomposition is orthogonal, the invariants emerge in a natural and systematic way. Their independence and completeness follow straightforwardly from the direct sum decomposition of the tensor space. For arbitrary anisotropic materials, we defined the Cauchy factor as a dimensionless measure of a closeness to a pure Cauchy material. Quite similarly, we defined isotropy factor as a measure for a closeness to an isotropic prototype of a given material. The irreducible factors are defined in order to characterize the contributions of different irreducible parts of an anisotropic elasticity tensor. This formalism can be useful for various elasticity problems: • Elasticity wave propagation [2]; • complete set of anisotropy invariants, see [28]; • material symmetries and wavefront symmetries [5], [6]; • averaging of anisotropic material by a higher symmetric prototype [18], [21].

Acknowledgments I would like to thank F.-W.Hehl (Cologne/Columbia, MO) and A. Norris (Rutgers) for most helpful discussion and comments.

A A.1

Calculating the relations between invariants The first Ting invariant

For the first invariant of Ting, we write B1 = C ijkl Cijkl =

5 X

(I) ijkl

5 X

(J)

Cijkl .

(142)

B1 = (1) S 2 + (2) S 2 + (3) S 2 + (1) A2 + (2) A2

(143)

I=1

C

J=1

Due to the orthogonality of the decomposition, we are left with

We calculate step by step 2 1 1 2 ij kl 1 (1) 2 S = S g g + g ik g jl + g il g jk = S 2 = Z1 , (144) 225 5 5 2 1 6 6 (2) 2 S = P ij g kl + P ik g jl + P il g jk + P jk g il + P jl g ik + P kl g ij = P ij Pij = Z4 , (145) 49 7 7 (3) 2 ijkl S = R Rijkl = Z7 , (146) (147) and (1)

A2 =

(2)

A2 =

2 1 2 ij kl A 2g g − g il g jk − g ik g jl = A2 = Z3 (148) 36 1 ikm jln + ilm jkn ikp jlq + ilp jkq Qmn Qpq = 12Qmn Qmn = 12Z6 . (149) 4

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21

Consequently, the first invariant of Ting reads 1 6 B1 = Z1 + Z3 + Z4 + 12Z6 + Z7 . 5 7

A.2

(150)

The second Ting invariant

For the second invariant of Ting, B2 , we first calculate the trace C i ikl = g ij Cijkl = g ij Sijkl + (1)Aijkl + (2)Aijkl .

(151)

Here, 1 g ij Sijkl = Skl = Pkl + Sgkl , 3

(152)

g ij (1)Aijkl = g ij (gik Qjl + gjk Qil + gil Qjk + gjl Qik − 2gkl Qij − 2gij Qkl ) = −2Qkl , and

2 1 g ij (2)Aijkl = Ag ij (2gij gkl − gil gjk − gik gjl ) = Agkl . 6 3

(153) (154)

Hence, C i ikl = Pkl − 2Qkl +

1 (S + 2A) gkl . 3

(155)

Consequently, the second invariant of Ting reads B2 = C i ikl Cj jkl = Pkl P kl − 4Pkl Qkl + 4Qkl Qkl + or

A.3

1 (S + 2A)2 , 3

1 4 4 B2 = Z1 + Z2 + Z3 + Z4 − 4Z5 + 4Z6 . 3 3 3

(156)

(157)

The first Ahmad invariant

For the first invariant of Ahmad, B3 , we need the trace (1) (2) j ij ij C kjl = g Cikjl = g Sikjl + Aikjl + Aikjl .

(158)

Here,

g ij (1)Aikjl

1 g ij Sikjl = Skl = Pkl + Sgkl , 3 1 1 = Ag ij (2gik gjl − gil gjk − gij gkl ) = − Agkl . 6 3

(159) (160)

and g ij (2)Aikjl = g ij (gij Qkl + gjk Qil + gil Qjk + gkl Qij − 2gjl Qik − 2gik Qjl ) = Qkl ,

(161)

Consequently, 1 C j kjl = Pkl + Qkl + gkl (S − A) . 3

(162)

Y. Itin


Hence using (155) and (162) we get 1 1 B3 = Pkl + Qkl + gkl (S − A) P kl − 2Qkl + (S + 2A) g kl 3 3 1 = Pkl P kl − Pkl Qkl − 2Qkl Qkl + (S + 2A)(S − A) , 3 or

1 1 2 B3 = Z1 + Z2 − Z3 + Z4 − Z5 − 2Z6 . 3 3 3

A.4

(163)

(164)

The second Ahmad invariant

This invariant is obtained by the use of the formula (162), 1 1 kl kl kl Pkl + Qkl + gkl (S − A) B3 = P + Q + g (S − A) 3 3 1 = Pkl P kl + 2Pkl Qkl + Qkl Qkl + (S − A)2 , 3 or

1 2 1 B4 = Z1 − Z2 + Z3 + Z4 + 2Z5 + Z6 . 3 3 3

A.5

22

(165)

(166)

The Norris invariant

We put the invariant of Norris first in the form B5 = C ijkl Cikjl = S ijkl + Aijkl (Sikjl + Aikjl ) = S ijkl Sijkl + Aijkl Aikjl .

(167)

Using (144,145,146), we have 1 6 S ijkl Sikjl = S ijkl Sijkl = Z1 + Z4 + Z7 . 5 7

(168)

We observe Aijkl Aikjl =

(1) ijkl

A

+ (2)Aijkl

Aikjl + (2)Aikjl = (1)Aijkl (1)Aikjl + (2)Aijkl (2)Aikjl .

(1)

(169)

Thus we find (1)

Aijkl (1)Aikjl = (2)

1 2 1 A 2gij gkl − gil gjk − gik gjl 2g ik g jl − g il g jk − g ij g kl = − A2 , 36 2

Aijkl (2)Aikjl =

(170)

gik Qjl + gjk Qil + gil Qjk + gjl Qik − 2gkl Qij − 2gij Qkl × g ij Qkl + g jk Qil + g il Qjk + g kl Qij − 2g jl Qik − 2g ik Qjl

= −6Qij Qij .

(171)

Consequently, 1 1 6 B5 = C ijkl Cikjl = Z1 − Z3 + Z4 − 6Z6 + Z7 . 5 2 7

(172)

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23

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[18] Moakher, M., Norris, A. N. (2006). The closest elastic tensor of arbitrary symmetry to an elasticity tensor of lower symmetry, Journal of Elasticity 85 (3), 215-263. [19] A. H. Nayfeh, Wave propagation in layered anisotropic media: with applications to composites (North-Holland, Amsterdam, 1995) [20] Norris, A. N. (2007) Quadratic invariants of elastic moduli, Quarterly J. Mech. Appl. Math. 60 (3), 367–389. [21] Norris, A.N., (2006) Elastic moduli approximation of higher symmetry for the acoustical properties of an anisotropic material, Journal of Acoustical Society of America 119 (4), 2114-2121 [22] P. Podio-Guidugli, (2000) A Primer in Elasticity, Kluwer, Dordrecht. [23] Sokolnikoff, I. S. (1956) Mathematical Theory of Elasticity, 2nd edn. New York: McGraw-Hill. [24] Y. Surrel, A new description of the tensors of elasticity based upon irreducible representations, Eur. J. Mech. A/Solids 12 (1993) 219–235. [25] T. C. T. Ting, (1987) Invariants of anisotropic elastic constants, Quarterly J. Mech. Appl. Math. 40 431–448. [26] J.H. Weiner, Statistical Mechanics of Elasticity (Dover, Mineola, New York, 2002) [27] Weyl, H. (1997) The classical groups: their invariants and representations Vol. 1, Princeton university press. [28] H. Xiao, (1998) On anisotropic invariants of a symmetric tensor: crystal classes, quasi-crystal classes and others, Proc. R. Soc. London A 454 1217–1240.